After a bit of experimenting with Demon Lord Ghirahim, we've come to grasp the basic structure of the friend request system:

If you wish to add someone as a friend on the forum, you may find the "Friends and Foes" tab as one of your options while on the page to edit your profile. After clicking on that tab, the option to send friend requests is the immediate option available on that page. Underneath it is the list of requests you have already sent, and below that is a list of any requests other have sent you. You can accept or deny friend requests from others there, as well as remove friends from your friends list if necessary. At the bottom of the page is your foes list, which serves as an ignore feature. It allows you to add users to your foes list and displays the ones currently on it.

For convenience, there's also a few links above the avatar images of users on their respective profile pages (titled "Add to my friends list" and "Add to my foes list"). This is a shortcut to the "Friends and Foes" profile tab and will send the add/ignore request when you click the "Add to my friends/foes list" link.

Note that friend requests sent by others can only be noticed on the "Friends and Foes" tab. In other words, you will not receive a private message or notification and must check it manually.